Curves and surfaces PDF
Preview Curves and surfaces
CurvesandSurfaces Marco Abate • Francesca Tovena Curves and Surfaces MarcoAbate FrancescaTovena DepartmentofMathematics DepartmentofMathematics UniversityofPisa(Italy) UniversityofRomeTorVergata (Italy) Translatedby: DanieleA.Gewurz DepartmentofMathematics,UniversityofRomeLaSapienza(Italy) TranslatedfromtheoriginalItalianedition: M.Abate,F.Tovena:Curveesuperfici,©Springer-VerlagItalia2006 UNITEXT–LaMatematicaperil3+2 ISSNprintedition:2038-5722 ISSNelectronicedition:2038-5757 ISBN978-88-470-1940-9 ISBN978-88-470-1941-6 (eBook) DOI10.1007/978-88-470-1941-6 LibraryofCongressControlNumber:2011934365 SpringerMilanDordrechtHeidelbergLondonNewYork ©Springer-VerlagItalia2012 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmsorinotherways,andstorageindata banks.Duplicationofthispublicationorpartsthereofispermittedonlyundertheprovi- sionsoftheItalianCopyrightLawinitscurrentversion,andpermissionforusemustalways beobtainedfromSpringer.ViolationsareliabletoprosecutionundertheItalianCopyright Law. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. 9 8 7 6 5 4 3 2 1 Coverdesign:BeatriceB,Milano Coverimage:inspiredby“Tetroidwith24Heptagons”byCarloH.Sequin TypesettingwithLATEX:PTP-Berlin,ProtagoTEX-ProductionGmbH,Germany (www.ptp-berlin.eu) PrintingandBinding:GrafichePorpora,Segrate(Mi) PrintedinItaly Springer-VerlagItaliaS.r.l.,ViaDecembrio28,I-20137Milano SpringerisapartofSpringerScience+BusinessMedia(www.springer.com) Preface This book is the story of a success. Classical Euclidean geometry has always had a weak point (as Greek ge- ometers were well aware): it was not able to satisfactorily study curves and surfaces, unless they were straight lines or planes. The only significant excep- tion were conic sections. But it was an exception confirming the rule: conic sections were seen as intersections of a cone (a set consisting of the lines through a point forming a constant angle with a given line) with a plane, andsotheycould bedealtwithusingthe lineargeometryoflinesandplanes. The theory of conic sections is rightly deemed one of the highest points of classical geometry; beyond it, darkness lay. Some special curves, a couple of very unusual surfaces; but of a general theory, not even a trace. Thefactis,classicalgeometersdidnothavethelanguage neededtospeak about curves or surfaces in general. Euclidean geometry was based axiomat- ically on points, lines, and planes; everything had to be described in those terms, while curves and surfaces in general do not lend themselves to be studied by means of that vocabulary. All that had to be very frustrating; it suffices to take a look around to see that our world is full of curves and surfaces, whereas lines and planes are just a typically human construction. ExeuntGreeksendEgyptians,centuriesgoby,Arabsbeginlookingaround, Italianalgebraistsbreakthebarrierofthirdandfourthdegreeequations,enter Descartes and discovers Cartesian coordinates. We are now at the beginning of 1600s, more than a millennium after the last triumphs of Greek geometry; at last, very flexible tools to describe curves and surfaces are available: they canbeseenasimagesorasvanishinglocioffunctionsgiveninCartesiancoor- dinates. The menagerie of special curves and surfaces grows enormously, and it becomes clear that in this historical moment the main theoretical problem is to be able to precisely define (and measure) how curves and surfaces differ from lines and planes. What does it mean that a curve is curved (or that a surface is curved, for that matter)? And how can we measure just how much a curve or a surface is curved? VI Preface Compared to the previous millennium, just a short wait is enough to get satisfactory answers to these questions. In the second half of 17th century, Newton and Leibniz discovered differential and integral calculus, redefining what mathematics is (and what the world will be, opening the way for the comingoftheindustrialrevolution).Newton’sandLeibniz’scalculusprovides effective tools to study, measure, and predict the behavior of moving objects. The path followed by a point moving on a plane or in space is a curve. The point moves along a straight line if and only if its velocity does not change direction; the more the direction of its velocity changes, the more its path is curved. So it is just natural to measure how much curved is the path by measuring how much the direction of the velocity changes; and differential calculus was born exactly to measure variations. Voila`, we have an effective andcomputabledefinitionofthecurvatureofacurve:thelengthoftheaccel- eration vector (assuming, as we may always do, that the point moves along the curve at constant scalar speed). The development of differential geometry of curves and surfaces in the following two centuries is vast and impetuous. In 18th century many gifted mathematicians already applied successfully to geometry the new calculus techniques, culminating in the great accomplishments of 19th century French school, the so-called fundamental theorems of local theory of curves and sur- faces,whichprovethatthenewlyintroducedinstrumentsare(necessaryand) sufficient to describe all local properties of curves and surfaces. Well, at least locally. Indeed, as in any good success story, this is just the beginning; weneedacoup de th´eatre.Thetheorywehavebrieflysummarized works very well for curves; not so well for surfaces. Or, more precisely, in the case of surfaces we are still at the surface (indeed) of the topic; the local de- scription given by calculus techniques is not enough to account for all global propertiesofsurfaces.Veryroughlyspeaking,curves,evenonalargescale,are essentially straight lines and circles somewhat crumpled in plane and space; on the other hand, describing surfaces simply as crumpled portions of plane, whileusefultoperformcalculations,isundulyrestrictiveandpreventsusfrom understanding and working with the deepest and most significant properties of surfaces, which go far beyond than just measuring curvature in space. Probably the person whowasmostawareof this situation wasGauss,one ofthegreatest(ifnotthe greatest)mathematicianoffirsthalfof19thcentury. Withtwomasterlytheorems,Gausssucceededbothinshowinghowmuchwas still to be discovered about surfaces, and in pointing to the right direction to be followed for further research. Withthefirsttheorem,hisTheorema Egregium,Gaussshowedthatwhile, seen from inside, all curves are equal (an intelligent one-dimensional being fromwithinsuchaone-dimensionalworldwouldnotbeabletodecidewhether helivesinastraightlineorinacurve),thisisfarfromtrueforsurfaces:there is a kind of curvature (the Gaussian curvature, in fact) that can be measured from within the surface, and can differ for different surfaces. In other words, it is possible to decide whether Earth is flat or curved without leaving one’s Preface VII backyard: it is sufficient to measure (with sufficiently precise instruments...) the Gaussian curvature of our garden. Thesecondtheorem,calledGauss-Bonnet theorem sinceitwascompleted and extended by Pierre Bonnet, one of Gauss’s brightest students, disclosed that by just studying local properties one can fail to realize that deep and significant phenomena take place at a global level, having geometrical im- plications at a local level too. For instance, one of the consequences of the Gauss-Bonnet theorem is that no matter how much we deform in space a sphere, by stretching it (without breaking) so to locally change in an appar- entlyarbitrarywayitsGaussiancurvature,the integral of Gaussian curvature on the whole surface is constant: every local variation of curvature is neces- sarily compensated by an opposite variation somewhere else. Another important topic pointed to the interrelation between local and global properties: the study of curves on surfaces. One of the fundamental problems in classical differential geometry (even more so because of its prac- tical importance) is to identify the shortest curve (the geodesic) between two givenpointsonasurface.Evensimply provingthatwhenthepoints areclose to each other there exists a unique geodesic joining them is not completely trivial; if the two points are far from each other, infinitely many geodesics could exist, or none at all. This is another kind of phenomenon of a typically global nature: it cannot be studied with differential calculus techniques only, which are inherently local. Once more, to go ahead and study global properties a new language and new tools were needed. So we arrive at the twentieth century and at the cre- ation and development (by Poincar´e and others) of topology, which turns out to be the perfect environment to study and understand global properties of surfaces. To give just one example, the description given by Hopf and Rinow of the surfaces in which all geodesics can be extended for all values of the time parameter (a property implying, in particular, that each pair of points is connected by a geodesic) is intrinsically topological. And this is just the beginning of an even more important story. Start- ing from seminal insights by Riemann (who, in particular, proved that the so-called non-Euclidean geometries were just geometries on surfaces different from the plane and not necessarily embedded in Euclidean space), definitions andideasintroducedtostudytwo-dimensionalsurfaceshavebeenextendedto n-dimensional manifolds, the analogous of surfaces in any number of dimen- sions.Differentialgeometryofmanifoldshasturnedouttobeoneofthemost significantareasincontemporarymathematics;itslanguageanditsresultsare used more or less everywhere (and not just in mathematics: Einstein’s gen- eraltheoryofrelativity, tomentionjustone example,couldnotexistwithout differentialgeometry).Andwhoknowswhatthefuture hasinstoreforus... This book’s goal is to tell the main threads of this story from the view- point of contemporary mathematics. Chapter 1 describes the local theory of curves, from the definition of what a curve is to the fundamental theorem of the local theory of curves. Chapters 3 and 4 deal with the local theory VIII Preface of surfaces, from the (modern) definition of a surface to Gauss’s Theorema Egregium. Chapter 5 studies geodesics on a surface, up to the proof of their existence and local uniqueness. The remaining chapters (and some of the chapters’ supplementary mate- rial; see below) are devoted instead to global properties. Chapter 2 discusses some fundamental results in the global theory of plane curves, including the Jordan curve theorem, which is a good example of the relatively belated de- velopment of the interest in global properties of curves: the statement of the Jordan curve theorem seems obvious until you try to actually prove it — and then it turns out to be surprisingly difficult, and far deeper than expected. Chapter6isdevotedtotheGauss-Bonnettheorem,withacompleteproof is given, and to some of its innumerable applications. Finally, in Chapter 7, we discuss a few important results about the global theory of surfaces (one of them, the Hartman-Nirenberg theorem, proved only in 1959), mainly focus- ingonthe connectionsbetweenthe signofGaussiancurvature and theglobal (topological and differential) structure of surfaces. We shall confine ourselves to curves and surfaces in the plane and in space; but the language and the methods we shall introduce are compatible with those used in differential ge- ometry of manifolds of any dimension, and so this book can be useful as a training field before attacking n-dimensional manifolds. Wehaveattemptedtoprovideseveralpossiblepathsforusingthisbookas a textbook. A minimal path consists in just dealing with local theory: Chap- ters 1, 3, and 4 can be read independently of the remaining ones, and can be used to offer, in a two-month course, a complete route from initial defini- tions to Gauss’s Theorema Egregium; if time permits, the first two sections of Chapter 5 can be added, describing the main properties of geodesics. Such a course is suitable and useful both for Mathematics (of course) and Physics students,fromthesecondyearon,andforEngineeringandComputerScience students (possibly for a Master degree) who need (or are interested in) more advanced mathematical techniques than those learnt in a first-level degree. In a one-semester course, on the other hand, it is possible to adequately cover the global theory too, introducing Chapter 2 about curves, the final section of Chapter 5 about vector fields, and, at the professor’s discretion, Chapter 6 about the Gauss-Bonnet theorem or Chapter 7 about the classifi- cationofclosedsurfaceswithconstantGaussiancurvature.Inourexperience, in a one-semester course given to beginning or intermediate undergraduate Mathematics or Physics students, it is difficult to find time to cover both topics, so the two chapters are completely independent; but in the case of an one-year course, or in an one-semester course for advanced students, it might be possible to do so, possibly even touching upon some of the supplementary material (see below). Eachchaptercomeswithseveralguidedproblems,thatis,solvedexercises, both computational (one of the nice features of local differential geometry of curves and surfaces is that almost everything can be explicitly computed — well, with the exception of geodesics) and of a more theoretical character, Preface IX whose goal is to teach you how to effectively use the techniques given in the corresponding chapter. You will also be able to test your skill by solving the large number of exercises we propose, subdivided by topic. You might have noticed another feature peculiar to this book: we address youdirectly,dearreader.Thereisaprecisereasonforthis:wewantyoutofeel activelyinvolvedwhilereadingthebook.Amathematicstextbook,nomatter at what level, is a sequence of arguments, exposed one after another with (hopefully) impeccable logic. While reading, we are led by the argument, up toapointwherewehavenoideawhytheauthorfollowedthatpathinsteadof another, and—evenworse—wearenotable toreconstructthat pathonour own. In order to learn mathematics, it is not enough to read it; you must do mathematics. The style we adopted in this book is chosen to urge you in this direction; in addition to motivations for each notion we shall introduce, you will often find direct questions trying to stimulate you to an active reading, without accepting anything just because you trust us (and, with any luck, they will also help you to stay awake, should you happen to study in the wee hours of the morning, the same hours we used to write this book...). This book also has the (vain?) ambition not to be just a textbook, but something more. This is the goal of the supplementary material. There is a wealth of extremely interesting and important results that does not usually find its place in courses (mostly due to lack of time), and that is sometimes hard to find in books. The supplementary material appended to each chapter displayachoice(suggested,asisnatural,byourtaste)oftheseresults.Wego fromcompleteproofsoftheJordancurvetheorem,alsoforcurvesthatareno morethancontinuous, andofthe existenceoftriangulations onsurfaces(the- oremsofteninvokedandusedbutveryrarelyproved),toadetailedexposition oftheHopf-RinowtheoremaboutgeodesicsorBonnet’sandHadamard’sthe- orems about surfaces with Gaussian curvature having a well-defined sign. We shall also prove that all closed surfaces are orientable, and the fundamental theorem of the local theory of surfaces (which, for reasons made clear at the endofChapter4,isnotusuallyincludedinastandardcurriculum),andmuch more. We hope that this extra material will answer questions you might have asked yourself while studying, arouse further you curiosity, and provide mo- tivation and examples for moving towards the study of differential geometry in any dimension. A warning: with just some exceptions, the supplementary materialissignificantlymorecomplicatedthantherestofthebook,andtobe understood it requires a good deal of participation on your part. A reassur- ance: nothing of what is presented in the supplementary material is used in the main part of the book. On a first reading, the supplementary results may be safely ignored with no detriment to the comprehension of the rest of the book.Lastly,mainlyduetospaceconsiderations,thesupplementarymaterial includes only a limited number of exercises. Twowordsaboutthenecessaryprerequisites.Asyoucanimagine,weshall use techniques and ideas from differential and integral calculus in several real variables,andfromgeneraltopology.Thenecessarynotionsfromtopologyare X Preface reallybasic:opensets,continuousfunctions,connectednessandcompactness, and even knowing those in metric spaces is enough. Since these topics are usually covered in any basic Geometry course, and often in Calculus courses too,wedidnotfeeltheneedtoprovidespecificreferencesforthefewtopology theorems we shall use. If necessary, you will find all of the above (and much more) in the first four chapters of [11]. Muchmorecarehasbeenputingivingtheresultsincalculusweshallneed, both because they typically are much deeper than those in general topology, and to give you statements consistent with what we need. All of them are standard results, covered in any Advanced Calculus course (with the possible exception of Theorem 4.9.1); a good reference text is [5]. On the other hand, we do not require you to have been exposed to al- gebraic topology (in particular, if you do not have any idea of what we are talkingabout,youdonothavetoworry).Forthisreason,Section2.1contains a complete introduction to the theory of the degree for continuous maps of the circle to itself, and Section 7.5 discusses everything we shall need about the theory of covering maps. Needless to say, this is not the first book about this topic, and it could not have been written ignoring the earlier books. We found especially useful the classical texts by do Carmo [4] and Spivak [22], and the less classical but not less useful ones by Lipschutz [13] and Montiel and Ros [16]; if you will like this book, you might want to have a look at those books as well. On theotherhand,goodstartingpointsforstudyingdifferentialgeometryinany dimension are the already cited [22], and [10] and [12]. Lastly, the pleasant duty of thanks. This book would never have seen the light of the day, and would certainly have been much worse, without the help, assistance, understanding, and patience of (in alphabetical order) Luigi Ambrosio, Francesca Bonadei, Piermarco Cannarsa, Cinzia Casagrande, Ciro Ciliberto, Michele Grassi, Adele Manzella, and Jasmin Raissy. We are partic- ularly grateful to Daniele A. Gewurz, who flawlessly completed the daunting task of translating our book from Italian to English. A special thanks goes to our students of all these years, who put up with several versions of our lec- ture notes and have relentlessly pointed out even the smallest error. Finally, a very special thanks to Leonardo, Jacopo, Niccolo`, Daniele, Maria Cristina, and Raffaele, who have fearlessly suffered the transformation of their parents in an appendix to the computer keyboard, and that with their smiles remind us that the world still deserved to be lived. Pisa and Rome, July 2011 Marco Abate Francesca Tovena
The list of books you might like
